(i) All questions are compulsory.

(ii) The question paper consists of 34 questions divided into four sections A, B, C and D.

(iii) Sections A contains 8 questions of one mark each, which are multiple choice type questions, section B contains 6 questions of two marks each, section C contains 10 questions of three marks each, and section D contains 10 questions of four marks each.

(iv) Use of calculators is not permitted.

Q1 :

The first three terms of an AP respectively are 3*y* – 1, 3*y* + 5 and 5*y* + 1. Then *y* equals:

(A) –3

(B) 4

(C) 5

(D) 2

**Answer :**

The first three terms of an AP are

We need to find the value of y.

We know that if a, b and c are in AP, then:

b − a = c − b ⇒ 2b = a + c

Hence, the correct option is C.

Q2 :

In Fig. 1, QR is a common tangent to the given circles, touching
externally at the point T. The tangent at T meets QR at P. If PT
= 3.8 cm, then the length of QR (in cm) is :

(A) 3.8

(B) 7.6

(C) 5.7

(D) 1.9

**Answer :**

It is known that the length of the tangents drawn from an external
point to a circle are equal.

∴ QP = PT = 3.8
cm
...(1)

PR = PT = 3.8
cm
...(2)

From equations (1) and (2), we get:

QP = PR = 3.8 cm

Now, QR = QP + PR

= 3.8 cm + 3.8 cm

= 7.6 cm

Hence, the correct option is B.

Q3 :

In Fig. 2, PQ and PR are two tangents to a circle with centre O.
If ∠QPR = 46°, then ∠QOR equals:

(A) 67°

(B) 134°

(C) 44°

(D) 46°

**Answer :**

Given: ∠QPR = 46°

PQ and PR are tangents.

Therefore, the radius drawn to these tangents will be perpendicular
to the tangents.

So, we have OQ ⊥ PQ and OR ⊥ RP.

⇒ ∠OQP = ∠ORP = 90^{∘}

So, in quadrilateral PQOR, we have

∠OQP +∠QPR + ∠PRO + ∠ROQ = 360^{∘}

⇒ 90° + 46° + 90° + ∠ROQ = 360^{∘}

⇒ ∠ROQ = 360^{∘} −
226^{∘} = 134^{∘}

Hence, the correct option is B.

Q4 :

A ladder makes an angle of 60° with the ground when placed
against a wall. If the foot of the ladder is 2 m away from the
wall, then the length of the ladder (in metres) is:

(A)

(B)

(C)

(D) 4

**Answer :**

Q5 :

If two different dice are rolled together, the probability of
getting an even number on both dice, is:

(A)

(B)

(C)

(D)

**Answer :**

Q6 :

A number is selected at random from the numbers 1 to 30. The
probability that it is a prime number is:

(A)

(B)

(C)

(D)

**Answer :**

Q7 :

If the points A(*x*, 2), B(−3, −4)
and C(7, − 5) are collinear, then the value of *x* is:

(A) −63

(B) 63

(C) 60

(D) −60

**Answer :**

Q8 :

The number of solid spheres, each of diameter 6 cm that can be
made by melting a solid metal cylinder of height 45 cm and
diameter 4 cm, is:

(A) 3

(B) 5

(C) 4

(D) 6

**Answer :**

Q9 :

Solve the quadratic equation 2*x*^{2} + *ax* − *a*^{2} = 0 for *x*.

**Answer :**

Q10 :

The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.

**Answer :**

Q11 :

Prove that the line segment joining the points of contact of two parallel tangents of a circle, passes through its centre.

**Answer :**

Q12 :

If from an external point P of a circle with centre O, two tangents PQ and PR are drawn such that ∠QPR = 120°, prove that 2PQ = PO.

**Answer :**

Q13 :

Rahim tosses two different coins simultaneously. Find the probability of getting at least one tail.

**Answer :**

Q14 :

In fig. 3, a square OABC is inscribed in a quadrant OPBQ of a
circle. If OA = 20 cm, find the area of the shaded region.
(Use π = 3.14)

**Answer :**

Q15 :

Solve the equation *x*.

**Answer :**

Q16 :

If the seventh term of an AP is ^{rd} term.

**Answer :**

Q17 :

Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle.

**Answer :**

Q18 :

If the point A(0, 2) is equidistant from the points B(3,
*p*) and C(*p*, 5), find
*p*. Also find the length of AB.

**Answer :**

Q19 :

Two ships are there in the sea on either side of a light house in
such a way that the ships and the light house are in the same
straight line. The angles of depression of two ships as observed
from the top of the light house are 60° and 45°. If the
height of the light house is 200 m, find the distance between the
two ships. [Use

**Answer :**

Q20 :

If the points A(−2, 1), B(*a*, *b*) and C(4, −1) are collinear and *a* − *b* = 1, find the values of
*a* and *b*.

**Answer :**

Q21 :

In Fig 4, a circle is inscribed in an equilateral triangle
ABC of side 12 cm. Find the radius of inscribed circle and the
area of the shaded region. [Use π = 3.14 and

**Answer :**

Q22 :

In Fig.5, PSR, RTQ and PAQ are three semicircles of diameters 10
cm, 3 cm and 7 cm respectively. Find the perimeter of the shaded
region. [Use π = 3.14]

**Answer :**

Q23 :

A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank which is 10 m in diameter and 2 m deep. If the water flows through the pipe at the rate of 4 km per hour, in how much time will the tank be filled completely?

**Answer :**

Q24 :

A solid metallic right circular cone 20 cm high and whose
vertical angle is 60°, is cut into two parts at the middle
of its height by a plane parallel to its base. If the frustum so
obtained be drawn into a wire of diameter

**Answer :**

Q25 :

The difference of two natural numbers is 5 and the difference of
their reciprocals is

**Answer :**

Q26 :

Prove that the length of the tangents drawn from an external point to a circle are equal.

**Answer :**

Q27 :

The angles of elevation and depression of the top and the bottom of a tower from the top of a building, 60 m high, are 30° and 60° respectively. Find the difference between the heights of the building and the tower and the distance between them.

**Answer :**

Q28 :

A bag contains cards numbered from 1 to 49. A card is drawn from
the bag at random, after mixing the cards thoroughly. Find the
probability that the number on the drawn card is:

(i) an odd number

(ii) a multiple of 5

(iii) a perfect square

(iv) an even prime number

**Answer :**

Q29 :

Find the ratio in which the point P(*x*, 2)
divides the line segment joining the points A(12, 5) and B(4,
– 3). Also find the value of *x*.

**Answer :**

Q30 :

Find the values of *k* for which the quadratic
equation (*k* + 4) *x*^{2} + (*k* + 1) *x* + 1 =
0 has equal roots. Also find these roots.

**Answer :**

Q31 :

In an AP of 50 terms, the sum of first 10 terms is 210 and the sum of its last 15 terms is 2565. Find the A.P.

**Answer :**

Q32 :

Prove that a parallelogram circumscribing a circle is a rhombus.

**Answer :**

Q33 :

Sushant has a vessel, of the form of an inverted cone, open at
the top, of height 11 cm and radius of top as 2.5 cm and is full
of water. Metallic spherical balls each of diameter 0.5 cm are
put in the vessel due to which

**Answer :**

Q34 :

From a solid cylinder of height 2.8 cm and diameter 4.2 cm, a
conical cavity of the same height and same diameter is hollowed
out. Find the total surface area of the remaining solid.

**Answer :**

- 10th Maths Paper Solutions Set 2 : CBSE Delhi Previous Year 2015
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